We study wildly ramified G-Galois covers $ϕ:Y \to X$ branched at B (defined over an algebraically closed field of characteristic p). We show that curves Y of arbitrarily high genus occur for such covers even when G, X, B and the inertia groups are fixed. The proof relies on a Galois action on covers of germs of curves and formal patching. As a corollary, we prove that for any nontrivial quasi-p group G and for any sufficiently large integer $σ$ with $p \nmid σ$, there exists a G-Galois étale cover of the affine line with conductor $σ$ above the point $\infty$.
We compute the ramification filtration on wildly ramified $p^2$-cyclic extensions of local fields of characteristic $p$. The ramification filtration on the compositum of two $p$-cyclic and $p^2$-cyclic extensions are also computed. As an application, some partial results towards Abhyankar's Inertia conjecture has been proved.
Given a Galois cover of curves over $\mathbb{F}_p$, we relate the $p$-adic valuation of epsilon constants appearing in functional equations of Artin L-functions to an equivariant Euler characteristic. Our main theorem generalises a result of Chinburg from the tamely to the weakly ramified case. We furthermore apply Chinburg's result to obtain a `weak' relation in the general case. In the Appendix, we study, in this arbitrarily wildly ramified case, the integrality of $p$-adic valuations of epsilon constants.
In order to study $p$-adic étale cohomology of an open subvariety $U$ of a smooth proper variety $X$ over a perfect field of characteristic $p>0$, we introduce new $p$-primary torsion sheaves. It is a modification of the logarithmic de Rham-Witt sheaves of $X$ depending on effective divisors $D$ supported in $X-U$. Then we establish a perfect duality between cohomology groups of the logarithmic de Rham-Witt cohomology of $U$ and an inverse limit of those of the mentioned modified sheaves. Over a finite field, the duality can be used to study wild ramification class field theory for the open su…
We revisit a statement of Birch that the field of moduli for a marked three-point ramified cover is a field of definition. Classical criteria due to Dèbes and Emsalem can be used to prove this statement in the presence of a smooth point, and in fact these results imply more generally that a marked curve descends to its field of moduli. We give a constructive version of their results, based on an algebraic version of the notion of branches of a morphism and allowing us to extend the aforementioned results to the wildly ramified case. Moreover, we give explicit counterexamples for singular curve…
Corvaja and Zannier conjectured that an abelian variety over a number field satisfies a modified version of the Hilbert property. We investigate their conjecture for products of elliptic curves using Kawamata's structure result for ramified covers of abelian varieties, and Faltings's finiteness theorem for rational points on higher genus curves.
To every double cover ramified in two points of a general trigonal curve of genus g, one can associate an étale double cover of a tetragonal curve of genus g+1. We show that the corresponding Prym varieties are canonically isomorphic as principally polarized abelian varieties.
We derive a closed-form expression for all genus 1 Hurwitz numbers, and give a simple new graph-theoretic interpretation of Hurwitz numbers in genus 0 and 1. (Hurwitz numbers essentially count irreducible genus g covers of the sphere, with arbitrary specified branching over one point, simple branching over other specified points, and no other branching. The problem is equivalent to counting transitive factorisations of permutations into transpositions.) These results prove a conjecture of Goulden and Jackson, and extend results of Hurwitz and many others.
Let $k$ be a complete nonarchimedean field and let $X$ be an affinoid closed disc over $k$. We classify the tamely ramified twisted forms of $X$. Generalizing work of P. Russell on inseparable forms of the affine line we construct explicit families of wildly ramified forms of $X$. We compute the class group and the Grothendieck group of forms of $X$ in certain cases.
In this paper, we give a simple description of the deformations of a map between two smooth curves with partially prescribed branching, in the cases that both curves are fixed, and that the source is allowed to vary. Both descriptions work equally well in the tame or wild case. We then apply this result to obtain a positive-characteristic Brill-Noether-type result for ramified maps from general curves to the projective line, which even holds for wild ramification indices. Lastly, in the special case of rational functions on the projective line, we examine what we can say as a result about fami…
Let $k$ be a field of characteristic 0, let $C/k$ be a uniquely trigonal genus 4 curve, and let $P \in C(k)$ be a simply ramified point of the uniquely trigonal morphism. We construct an assignment of an orbit of an algebraic group of type $E_8$ acting on a specific variety to each element of $J_C(k)/2$. The algebraic group and variety are independent of the choice of $(C,P)$. We also construct a similar identification for uniquely trigonal genus 4 curves $C$ with $P \in C(k)$ a totally ramified point of the trigonal morphism. Our assignments are analogous to the assignment of a genus 3 curve …
We study the ramified Prym map $\mathcal P_{g,r} \longrightarrow \mathcal A_{g-1+\frac r2}^δ$ which assigns to a ramified double cover of a smooth irreducible curve of genus $g$ ramified in $r$ points the Prym variety of the covering. We focus on the six cases where the dimension of the source is strictly greater than the dimension of the target giving a geometric description of the generic fibre. We also give an explicit example of a totally geodesic curve which is an irreducible component of a fibre of the Prym map ${\mathcal P}_{1,2}$.
We construct automorphic representations for quasi-split groups $G$ over the function field $F=k(t)$ one of whose local components is an epipelagic representation in the sense of Reeder and Yu. We also construct the attached Galois representations under the Langlands correspondence. These Galois representations give new classes of conjecturally rigid, wildly ramified $^{L}G$-local systems over $\mathbb{P}^{1}-\{0,\infty\}$ that generalize the Kloosterman sheaves constructed earlier by Heinloth, Ngô and the author. We study the monodromy of these local systems and compute all examples when $G$ …
For any nonconstant f,g in C(x) such that the numerator H(x,y) of f(x)-g(y) is irreducible, we compute the genus of the normalization of the curve H(x,y)=0. We also prove an analogous formula in arbitrary characteristic when f and g have no common wildly ramified branch points, and generalize to (possibly reducible) fiber products of nonconstant morphisms of curves f:A-->D and g:B-->D.
Let $ϕ:\,X\rightarrow Y$ be a (possibly ramified) cover between two algebraic curves of positive genus. We develop tools that may identify the Prym variety of $ϕ$, up to isogeny, as the Jacobian of a quotient curve $C$ in the Galois closure of the composition of $ϕ$ with a well-chosen map $Y\rightarrow \mathbb{P}^1$. This method allows us to recover all previously obtained descriptions of a Prym variety in terms of a Jacobian that are known to us, besides yielding new applications. We also find algebraic equations for some of these new cases, including one where $X$ has genus $3$, $Y$ has genu…
In this paper we prove that the Prym map, from the space of double coverings of a curve of genus g with r branch points to the moduli space of abelian varieties, is generically injective if r>6 and g>1, r=6 and g>2, r=4 and g>4, r=2 and g>5. We also show that a very generic Prym variety of dimension at least 4 is not isogenous to a Jacobian.
Inside the moduli space of curves of genus 2 with 2 marked points we consider the loci of curves admitting a map to P^1 of degree d totally ramified over the two marked points, for d>= 2. Such loci have codimension two. We compute the class of their compactifications in the moduli space of stable curves. Several results will be deduced from this computation.
A general smooth curve of genus six lies on a quintic del Pezzo surface. In \cite{AK11}, Artebani and Kondō construct a birational period map for genus six curves by taking ramified double covers of del Pezzo surfaces. The map is not defined for special genus six curves. In this paper, we construct a smooth Deligne-Mumford stack $\mathfrak{P}_0$ parametrizing certain stable surface-curve pairs which essentially resolves this map. Moreover, we give an explicit description of pairs in $\mathfrak{P}_0$ containing special curves.
We give explicit computational algorithms to construct minimal degree (always $\le 4$) ramified covers of $\Prj^1$ for algebraic curves of genus 5 and 6. This completes the work of Schicho and Sevilla (who dealt with the $g \le 4$ case) on constructing radical parametrisations of arbitrary genus $g$ curves. Zariski showed that this is impossible for the general curve of genus $\ge 7$. We also construct minimal degree birational plane models and show how the existence of degree 6 plane models for genus 6 curves is related to the gonality and geometric type of a certain auxiliary surface.
In this paper we find an explicit formula for the number of topologically different ramified coverings $C\to\CP^1$ (C is a compact Riemann surface of genus g) with only one complicated branching point in terms of Hodge integrals over the moduli space of genus g curves with marked points.
Let $C$ be a curve of genus $g\geqslant 2$ defined over the fraction field $K$ of a complete discrete valuation ring $R$ with algebraically closed residue field. Suppose that $\char(K)=0$ and that the characteristic of the residue field is not 2. Suppose that the Jacobian $\Jac(C)$ has semi-stable reduction over $R$. Embed $C$ in $\Jac(C)$ using a $K$-rational point. We show that the coordinates of the torsion points lying on $C$ lie in the unique moderately ramified quadratic extension of the field generated over $K$ by the coordinates of the $p$-torsion points on $\Jac(C)$.
Let $k$ be an algebraically closed field of characteristic $p>0$. Let $G$ be $Z/\ell Z$ semi-direct product $Z/pZ$ where $\ell$ is a prime distinct from $p$. In this paper, we study Galois covers $ψ:Z \to P^1_k$ ramified only over $\infty$ with Galois group $G$. We find the minimal genus of a curve $Z$ that admits such a cover and show that it depends only on $\ell$, $p$, and the order $a$ of $\ell$ modulo $p$. We also prove that the number of curves $Z$ of this minimal genus which admit such a cover is at most $(p-1)/a$.
This article is an extended version of preprint math.AG/9902104. We find an explicit formula for the number of topologically different ramified coverings of a sphere by a genus g surface with only one complicated branching point in terms of Hodge integrals over the moduli space of genus g curves with marked points.
Faltings' theorem [Fal83],[Fal91] (formerly the Mordell conjecture [Mo22]) states that a curve of genus greater than one over any number field has only finitely many points. Again a natural question is how many points can such a curve have. Caporaso, Harris, and Mazur [CHM97] have shown that the weak Bombieri-Lang conjecture implies that for any number field $F$ and any integer $g \ge 2$ there is an absolute upper bound $B(F; g)$ on the number of points on a genus $g$ curve over $F$. Furthermore, the strong Bombieri-Lang conjecture implies that for each genus $g \ge 2$, there is an absolute bo…
Let $Y_K\to X_K$ be a ramified cyclic covering of curves, where $K$ is a cyclotomic field. In this work we study the $p$-rank of the reduction $\rm{mod}\, p$ of a model of the jacobian of $Y_K$. In this way, we obtain counterparts of the Deuring polynomial, defined for elliptic curves, for genus greater than one. Moreover, we show that curves $Y_K$ give Hecke characters for cyclotomic fields. To carry out this study we use Jacobi sums and certain $L$-functions.
We extend results on Weierstrass semigroups at ramified points of double covering of curves to any numerical semigroup whose genus is large enough. As an application we strengthen the properties concerning Weierstrass weights in \cited{To}.
In this paper we consider the Prym map for double coverings of curves of genus $g$ ramified at $r>0$ points. That is, the map associating to a double ramified covering its Prym variety. The generic Torelli theorem states that the Prym map is generically injective as soon as the dimension of the space of coverings is less or equal to the dimension of the space of polarized abelian varieties. We prove the generic injectivity of the Prym map in the cases of double coverings of curves with: (a) $g=2$, $r=6$, and (b) $g= 5$, $r=2$. In the first case the proof is constructive and can be extended to …
We study Shimura curves of PEL type in $\mathsf{A}_g$ generically contained in the Prym locus. We study both the unramified Prym locus, obtained using étale double covers, and the ramified Prym locus, corresponding to double covers ramified at two points. In both cases we consider the family of all double covers compatible with a fixed group action on the base curve. We restrict to the case where the family is 1-dimensional and the quotient of the base curve by the group is $\mathbb{P}^1$. We give a simple criterion for the image of these families under the Prym map to be a Shimura curve. Usin…
Let $ε>0$. In this article we will present a deterministic algorithm which does the following. The input is a hyperelliptic curve $C$ of genus $g$ over a finite field $k$ of cardinality $q$ given by $y^2+h(x)y=f(x)$ such that the $x$-coordinate map is ramified at $\infty$. In time $O(g^{2+ε} q^{1/2+ε})$ the algorithm outputs a set of generators of the Picard group $\mathrm{Pic}^0_k(C)$. This extends results which others have obtained when $g=1$. In this article we introduce a combinatorial tool, the `shape parameter', which we use together with character sum estimates from class field theory t…
{\it $(N,γ)$-hyperelliptic} semigroups were introduced by Fernando Torres to encapsulate the most salient properties of Weierstrass semigroups associated to totally-ramified points of $N$-fold covers of curves of genus $γ$. Torres characterized $(2,γ)$-hyperelliptic semigroups of maximal weight whenever their genus is large relative to $γ$. Here we do the same for $(3,γ)$-hyperelliptic semigroups, and we formulate a conjecture about the general case whenever $N \geq 3$ is prime.
We study "pure-cycle" Hurwitz spaces, parametrizing covers of the projective line having only one ramified point over each branch point. We start with the case of genus-0 covers, using a combination of limit linear series theory and group theory to show that these spaces are always irreducible. In the case of four branch points, we also compute the associated Hurwitz numbers. Finally, we give a conditional result in the higher-genus case, requiring at least 3g simply branched points. These results have equivalent formulations in group theory, and in this setting complement results of Conway-Fr…
Orders on surfaces provide a rich source of examples of noncommutative surfaces. Other than some existence results, very little is known about the various moduli spaces that can be associated to them. Even fewer examples have been explicitly computed. In this paper we compute the Picard and Hilbert schemes of an order on the projective plane ramified on a union of two conics. Our main result is that, upon carefully selecting the right Chern classes, the Hilbert scheme is a ruled surface over a genus two curve. Furthermore, this genus two curve is, in itself, the Picard scheme of the order.
In this thesis we have proved a conjecture about the moduli space SU_X(3) of semi-stable rank 3 vector bundles with trivial determinant on a genus 2 curve X, due to I. Dolgachev. Given X a smooth projective curve of genus 2, and the embedding of the jacobian JX into |3Θ|, A. Coble proved, at the begining of XX century, that there exists a unique cubic hypersurface C in |3Θ|* \simeq P^8, JX[3]-invariant and singular along JX. On the other hand, we have a map of degree 2 from SU_X(3) over |3Θ|, ramified along a sextic hypersurface B. The Dolgachev's conjecture affirms that the sextic B is the du…
In this paper we introduce a new invariant for the action of a finite group $G$ on a compact complex curve of genus $g$. With the aid of this invariant we achieve the classification of the components of the moduli space of curves with an effective action by the dihedral group $D_n$. This invariant has been used in the meanwhile by the authors in order to extend the genus stabilization result of Livingston and Dunfield and Thurston to the ramified case. This new version contains an appendix clarifying the correspondence between the above components and the image loci in the moduli space M_g (cl…
Branched covers of the complex projective line ramified over $0,1$ and $\infty$ (Grothendieck's {\em dessins d'enfant}) of fixed genus and degree are effectively enumerated. More precisely, branched covers of a given ramification profile over $\infty$ and given numbers of preimages of $0$ and $1$ are considered. The generating function for the numbers of such covers is shown to satisfy a PDE that determines it uniquely modulo a simple initial condition. Moreover, this generating function satisfies an infinite system of PDE's called the KP (Kadomtsev-Petviashvili) hierarchy. A specification of …
We study the components of the Hurwitz scheme of ramified coverings of $\mathbb{P}^1$ with monodromy given by the alternating group $A_6$ and elements in the conjugacy class of product of two disjoint cycles. In order to detect the connected components of the Hurwitz scheme, inspired by the case of the spin structures studied by Fried for the $3$-cycles, we use as invariant the lifting to the Valentiner group, triple covering of $A_6$. We prove that the Hurwitz scheme has two irreducible components when the genus of the covering is greater than zero, in accordance with the asymptotic solution …
There is a natural sequence of CY manifolds that are double covers of the projective g dimensional spaces ramified over 2g+2 hyperplanes. We observe that some of them are obtained as quotient of the action of the semi-direct product of g-1 copies of cyclic Z/2Z groups with the symmetric g group on the product of g-copies of hyper-elliptic curves of genus g. The quotient is a double cover of the projective g space ramified over 2g+2 hyperplanes. This construction generalizes the construction of a Kummer surface. The Kodaira-Spencer classes on the Jacobian are invariant under the action of the g…
A Beauville surface (of unmixed type) is a complex algebraic surface which is the quotient of the product of two curves of genus at least 2 by a finite group G acting freely on the product, where G preserves the two curves and their quotients by G are isomorphic to the projective line, ramified over three points. Such a group G is called a Beauville group. We show that if a characteristically simple group G is a cartesian power of a finite simple alternating group, then G is a Beauville group if and only if it has two generators and is not isomorphic to A_5.
The paper studies a relation between fundamental group of the complement to a plane singular curve and the orbifold pencils containing it. The main tool is the use of Albanese varieties of cyclic covers ramified along such curves. Our results give sufficient conditions for a plane singular curve to belong to an orbifold pencil, i.e. a pencil of plane curves with multiple fibers inducing a map onto an orbifold curve whose orbifold fundamental group is non trivial. We construct an example of a cyclic cover of the projective plane which is an abelian surface isomorphic to the Jacobian of a curve …
Let $k$ be an algebraically closed field of characteristic $p > 0$. We study obstructions to lifting to characteristic 0 the faithful continuous action $ϕ$ of a finite group $G$ on $k[[t]]$. To each such $ϕ$ a theorem of Katz and Gabber associates an action of $G$ on a smooth projective curve $Y$ over $k$. We say that the KGB obstruction of $ϕ$ vanishes if $G$ acts on a smooth projective curve $X$ in characteristic 0 in such a way that $X/H$ and $Y/H$ have the same genus for all subgroups $H \subset G$. We determine for which $G$ the KGB obstruction of every $ϕ$ vanishes. We also consider anal…
We consider multi-matrix models that are generating functions for the numbers of branched covers of the complex projective line ramified over $n$ fixed points $z_i$, $i=1,\dots,n$, (generalized Grotendieck's dessins d'enfants) of fixed genus, degree, and the ramification profiles at two points, $z_1$ and $z_n$. Ramifications at other $n-2$ points enter the sum with the length of the profile at $z_2$ and with the total length of profiles at the remaining $n-3$ points. We find the spectral curve of the model for $n=5$ using the loop equation technique for the above generating function represente…
A curve $X$ is said to be of type $(N,γ)$ if it is an $N$--sheeted covering of a curve of genus $γ$ with at least one totally ramified point. A numerical semigroup $H$ is said to be of type $(N,γ)$ if it has $γ$ positive multiples of $N$ in $[N,2Nγ]$ such that its $γ^{th}$ element is $2Nγ$ and $(2γ+1)N \in H$. If the genus of $X$ is large enough and $N$ is prime, $X$ is of type $(N,γ)$ if and only if there is a point $P \in X$ such that the Weierstrass semigroup at $P$ is of type $(N,γ)$ (this generalizes the case of double coverings of curves). Using the proof of this result and the Buchweitz…
We argue that for a smooth surface S, considered as a ramified cover over the projective plane branched over a nodal-cuspidal curve B one could use the structure of the fundamental group of the complement of the branch curve to understand other properties of the surface and its degeneration and vice-versa. In this paper, we look at embedded-degeneratable surfaces - a class of surfaces admitting a planar degeneration with a few combinatorial conditions imposed on its degeneration. We close a conjecture of Teicher on the virtual solvability of the mentioned fundamental group for these surfaces a…
Let E be the supersingular elliptic curve defined over k, the algebraic closure of the finite field with two elements, which is unique up to k-isomorphism. Denote by 0 its identity element and let C be the quotient of E-{0} under the action of the group Isom(E) (which is non-abelian, of order 24). The main result of this paper asserts that the set C(k) naturally parametrizes k-isomorphism classes of Lamé covers, which are tamely ramified covers of the projective line unramified outside three points having a particular ramification datum. This fact is surprising for two reasons: first of all, i…
